Lissajous Golden φ

Harmonographs are mechanical drawing devices invented in the 1840s that use coupled pendulums to trace Lissajous-like curves with slowly decaying amplitude. The mathematics is simple — x(t) = A·sin(f₁t + φ)·e^(−dt), y(t) = B·sin(f₂t)·e^(−dt) — but the visual result is anything but: looped rosettes, knotted braids, and spirographs that breathe inward as the pendulums lose energy. By choosing frequency ratios near small integer fractions, you get closed figures; choose them irrational and the curve fills the canvas with quasi-periodic embroidery. Lissajous figures, the closed limit case, were first studied by Jules-Antoine Lissajous in 1857 to visualize sound interference. Every image in this collection is the literal trace of two oscillators, rendered as continuous strokes through phase space.

x(t) = A sin(f₁t + φ)e^{−dt}, y(t) = B sin(f₂t)e^{−dt}